ESS 575 Models for Ecological Data

Dynamic Models: Forecasting Effects of Harvest on Lynx

April 24, 2019

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Problem

The Eurasian lynx (Lynx lynx) is a medium-sized predator with broad distribution in the boreal forests of Europe and Siberia. The lynx is classified as a threatened species throughout much of its range. There is controversy about legal harvest of lynx in Sweden. Proponents of harvest argue that allowing hunting of lynx reduces illegal kill (poaching). Moreover, Sweden is committed to regulate lynx numbers to prevent excessive predation on reindeer because reindeer are critical to the livelihoods of indigenous pastoralists, the Sami. Many environmentalists oppose harvest, however, arguing that lynx are too rare to remove their fully protected status. A similar controversy surrounds management of wolves in the Western United States.

Your task is to help resolve this controversy by developing a forecasting model for the abundance of lynx in a management unit in Sweden. You have data on the number of lynx family groups censused in the unit as well as annual records of the number of lynx harvested from the unit. You will model the population using the deterministic model: \[N_t=\lambda(N_{t-1}-H_{t-1}).\] where \(N_{t}\) is the true, unobserved abundance of lynx and \(H_{t-1}\) is the number of lynx harvested during \(t-1\) to \(t\). The parentheses in this expression reflect the fact that harvest occurs immediately after census, such that the next years population increment comes from the post-harvest population size. (For the population ecologists: What would be the model if harvest occurred immediately before census? Three months after census? Continuously throughout the year?)

*Immediately before census:

\[N_t=\lambda N_{t-1}-H_{t-1}.\] *Three months after census:

\[N_t=(\lambda^{\frac{1}{4}}N_{t-1}-H_{t-1})\lambda^{\frac{3}{4}}\] \[N_t=(\lambda^{\frac{1}{4}}N_{t-1}-H_{t-1})\lambda^{\frac{3}{4}}\] *Throughout the year: \[N_t=(\lambda^{\frac{1}{2}}N_{t-1}-H_{t-1})\lambda^{\frac{1}{2}}\] \[N_t=\lambda N_{t-1}-\lambda^{\frac{1}{2}}H_{t-1}\]

We can reasonably assume the harvest is observed without error. It may be a bit of a stretch to assume an that observations of \(N_t\) are made perfectly, but my Swedish colleagues have convinced me that their census method, snow tracking, does a good job of estimating the number of family groups (if not the number of lynx) in a management region.

The challenge in this problem is that the observations of lynx abundance (family groups) is not the same as the observation of harvest (number of lynx). Fortunately, you have prior information on the proportional relationship between number of family groups and number of lynx in the population, i.e, \[\phi=\frac{f}{N}\] where \(f\) is the number of family groups and \(N\) is the population size, mean \(\phi\)=.163 with standard deviation of the mean = .012.

1. Develop a hierarchical Bayesian model (also called a state space model) of the lynx population in the management unit. Diagram the Bayesian network of knowns and unknowns and write out the posterior and factored joint distribution.
   

2. Write JAGS code to approximate the marginal posterior distribution of the unobserved, true state over time (\(\mathbf{N}\)), the parameters in the model \(\lambda\) and \(\phi\) as well as the process variance and observation variance. Summarize the marginal posterior distributions of the parameters and unobserved states. Data are in the accompanying file, Lynx data new.csv.

3. Check MCMC chains for model parameters, process variance, and latent states for convergence. This will probably require using the excl option in MCMCsummary.

4. Conduct posterior predictive checks by simulating a new dataset for family groups (\(f_t\)) at every MCMC iteration. Calculate a Bayesian p value using the sums of squared discrepancy between the observed and the predicted number of family groups based on observed and simulated data, \[T^{observed} = \sum_{t=1}^{n}(f_{t}^{observed}-N_{t}\phi)^{2} \\T^{model} = \sum_{t=1}^{n}(f_{t}^{simulated}-N_{t}\phi)^{2}.\] The Bayesian p value is the proportion of MCMC iterations for which \(T^{model}>T^{obs}\).

5. Assure that your process model adequately accounts for temporal autocorrelation in the residuals, allowing the assumption that they are independent and identically distributed. To do this, include a derived quantity \[e_t=y_t-N_t\] in your JAGS code and JAGS object. Use the following code or something like it to examine how autocorrelation in the residuals changes with time lag. Write a paragraph describing how to interpret the plot produced by this function.

acf(unlist(MCMCpstr(z,param="e",func=mean)),main="", lwd = 3, ci=0)

The autocorrelation of time series is the Pearson correlation between values of the process at two different times, as a function of the lag between the two times. It can take on values between -1 and 1. Values close to 1 or -1 indicate a high degree of correlation. The residuals not auto correlated if their values drop close to 0 at relatively short lags and alternate between positive and negative values at subsequent lags. This plot reveals no autocorrelation in the residuals of our model.

6. Plot the median of the marginal posterior distribution of the number of lynx family groups over time (1998-2016) including a highest posterior density interval. Include your forecast for 2017 (the predictive process distribution) in this plot. Your plot should resemble Figure 1, below.

7. Decisions on the number of lynx to be harvested must be made before the population is censused, even though harvest occurs immediately after census. This is because it takes months to properly issue licenses to the designated number of hunters. Make a forecast of the number of family groups in 2018 assuming five alternative levels for 2017 harvest (0, 10, 25, 50, and 75 animals). Environmentalists and hunters have agreed on a acceptable range for lynx abundance in the unit, 26 - 32 family groups. Compute the probability that the post-harvest number of family groups will be below, within, and above this range during 2018. Tabulate these values. Hint: Set up a “model experiment” in your JAGS code where you forecast the number of lynx family groups during 2018 under the specified levels of harvest. Extract the MCMC chains for the forecasted family groups( e.g., fg.hat) using MCMCchains(coda_object, params = fg.hat) Use the ecdf function on the R side to compute the probabilities that the forecasted number groups will be below, within, or above the acceptable range.

A note about the data. Each row in the data file gives the observed number of family groups for that year in column 2 and that year’s harvest in column 3. The harvest in each row influences the population size in the next row. So, for example, the 2016 harvest influences the 2017 population size.

Use a lognormal distribution to model the true lynx population size over time. Use a Poisson distribution for the data model relating the true, unobserved state (the total population size) to the observed data (number of family groups).

An alternative approach, which is slightly more difficult to code, is to model the process as \(\text{negative binomia}(N_t|\lambda(N_{t-1}-H_{t-1}, \rho))\) and model the data as \(\text{binomial}(y_t|N_t,\phi)\). Explain why this second formulation might be better than the formulation you are using. (It turns out they give virtually identical results.)

There are two advantages to the negative binomial process model and binomial data model. A negative binomial process model treats the true state as an integer, which for small populations like this one, has some advantages because it includes demographic stochasticity. The binomial data model assures that the observed state is never larger than the true state, which makes sense if the only source of error in the census is failing to observe family that are present. On the other hand, if there is a possibility of double-counting, which is the case here, the then Poisson is a better choice for the data model.

Code

Preliminaries

library(rjags)

## Loading required package: coda

## Linked to JAGS 4.2.0

## Loaded modules: basemod,bugs

library(MCMCvis)
library(HDInterval)
y=read.csv("Lynx data new.csv")
# Levels of  Harvest to evaluate relative to goals for forecasting part.
h=c(0, 10, 25, 50, 75)
#Function to get beta shape parameters from moments
shape_from_stats <- function(mu = mu.global, sigma = sigma.global){
         a <-(mu^2-mu^3-mu*sigma^2)/sigma^2
         b <- (mu-2*mu^2+mu^3-sigma^2+mu*sigma^2)/sigma^2
        shape_ps <- c(a,b)
        return(shape_ps)
}

#get parameters for distribution of population multiplier, 1/p
shapes=shape_from_stats(.163,.012)
#check prior on p using simulated data from beta distribution
x = seq(0,1,.001)
p=dbeta(x,shapes[1],shapes[2])
plot(x,p,typ="l",xlim=c(.1,.3))

Simulate data for initial values and model verification

I almost always simulate the true state by choosing some biologically reasonable values for model parameters and “eyeballing” the fit of the true state to the data. I then use these simulated values for initial conditions as you can see in the inits list below. This is particularly important. Failing to give reasonable initial conditions for dynamic models can cause no end of problems in model fitting. Remember, you must have initial conditions for all unobserved quantities in the posterior distribution. It is easy to forget this because some of these quantities don’t have priors.

##visually match simulated data with observations for initial conditions
#visually match simulated data with observations for initial conditions
endyr = nrow(y)
n=numeric(endyr+1)
mu=numeric(endyr+1) #use this for family groups
lambda=1.1
sigma.p=.00001
n[1] = y$census[1]

for(t in 2: (endyr+1)){
    n[t] <- lambda*(y$census[t-1] - .16 * y$harvest[t-1])  #use this for family groups
    }
plot(y$year, y$census,ylim=c(0,100),xlab="Year", ylab="Population size", main="Simulated data")
lines(y$year,n[1:length(y$year)])

Initial values and data

#Data for JAGS
data = list(
    y.endyr = endyr,
    y.a=shapes[1], 
    y.b=shapes[2],
    y.H=y$harvest,
    y=y$census,
    h=h
)

inits = list(
    list(
    lambda = 1.2,
    sigma.p = .01,
    N=n
    ),
    list(
    lambda = 1.01,
    sigma.p = .2,
    N=n*1.2),
    list(
    lambda = .95,
    sigma.p = .5,
    N=n*.5)
    )

## Compiling model graph
##    Resolving undeclared variables
##    Allocating nodes
## Graph information:
##    Observed stochastic nodes: 18
##    Unobserved stochastic nodes: 48
##    Total graph size: 1280
## 
## Initializing model

Figure 1. Median population size of lynx (solid line) during 1997-2016 and forecasts for 2017 with 95% credible intervals (dashed lines). Red dotted lines give acceptable range of number of family groups determined in public input process.